Schrodinger scale asch

Even within the asymptotic regime, (, oo), the scalet solutions may not approximate the actual Schrodinger equation solution sufBciently well. We define another scale, the Schrodinger scale, asch> where this is the case. 

First, we need to define the Schrodinger scale, asch b), at which the scalet solutions do approximate, closely, the corresponding Schrodinger solution. This scale is defined for any of the decaying scalet solutions, regardless of the physical or unphysical (unbounded) nature of the Schrodinger solution they are converging to. 

In general, it is difficult to develop a TPQ analysis within the scalet representation because the (approximate) conditions to be imposed (i.e zero kinetic energy) can only be done at scales, oq, that can be close to asch- At the Schrodinger scale asch-, the scalet equation solutions are close to the corresponding Schrodinger equation solution they are converging to, and the local structure of the TPQ conditions would be incapable of efficiently distinguishing between physical and unphysical solutions. 

We define the Schrodinger equation scale (at which the decaying scalet solutions become very close to the Schrodinger equation solution they are converging to), asch(b), to correspond to the scale at which the lowest order asymptotic correction is dominant. This definition is more sensitive to the slower convergence rate of the aforementioned, kinetic energy related, scaling transform derivatives. 

At asch b) the scalet solution comes close, pointwise, to the Schrodinger solution. However, already at scales close to Op, Uo a,Tys approximation to S(t) can be quite good (better than 1%). 

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